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120       XII!    INTEGRATION

then a function u e Jf R(X) such that/- u  0, we may also assume that/^ 0.
Now let N be the set of points x e X such that/(*) = + oo ; we have ncpN f
for all integers n > 0, hence >^*(<pN)  /^*(/), and the hypothesis therefore
implies that ju*(<pN) = 0.

The relation "/(*) = #(*) almost everywhere in X " is an equivalence
relation between mappings of X into a set E, because the union of two
negligible sets is negligible. We say then that / and g are equivalent (with
respect to u) or ^-equivalent, and we shall denote by / the equivalence class
of a mapping/: X -> E. A mapping/: A - E, where A is a subset of X, is
said (by abuse of language) to be defined almost everywhere in X if X  A is
negligible. The equivalence class of such a mapping /is then defined to be the
equivalence class of any mapping of X into E which extends /; as before,
we denote the equivalence class by /. Two functions / g defined almost
everywhere are said to be equivalent if/ = g ; this means that the set of points
x e X at which /and g are both defined and/(jc) = g(x) has a negligible set
for its complement.

(13.6.5)   Let f, g be two equivalent mappings ofX into R. Then /x*(/) = H*(g)-

Let N be the negligible set of points x e X such that /(jc) ^ g(x). Since
the functions /, g and sup(/, g) are equal on X  N, we may assume that
/<; g. Let h be the negligible function which is equal to + oo at the points of
N, and to 0 elsewhere. If v e J is such that/^ v, the function v + h is defined
at all points of X, and we have g :g v + h, so that

by virtue of (13.5.6), because /**(#) > -co. From the definition of /**(/), it
follows that ii*(g) ^ /**(/) and hence that /*%) = /**(/).

If /is a mapping into R which is defined and finite almost everywhere in
X, then /is equivalent to a finite function defined on the whole of X. If/ g
are functions defined almost everywhere in X, with values in K, and finite
almost everywhere, then the same is true of/+ g and/#, and the equivalence
classes of these functions depend only on /and g. They are denoted byf+g
and/*?, respectively.

lff(x) ^ g(x) almost everywhere, then/^*) g g^x) almost everywhere for
any functions /15 gl equivalent to /, g, respectively. In this case we write
/<* #, and this defines an order relation on the set of equivalence classes (with
respect to /*) of mappings of X into R.pact spaces and TT : X -> Y a proper continuous mapping